DFT+DMFT and its Discontents: towards a first-principles theory of correlated electron...

Department of PhysicsColumbia UniversityCopyright A. J. Millis 2013

DFT+DMFT and its Discontents:towards a first-principles theory of correlated

electron materials

A. J. Millis

Department of Physics, Columbia University

Support: US-DOE-ER-046169US-ARO-56032-PHUS-NSF-DMR-1006282

Hyowon Park, Andrew J. Millis, and Chris A. Marianetti, PRL 109, 156402 (2012)E. Gull, O. Parcollet and A. J. Millis, PRL 110 216406 (2013)H. T. Dang, A. J. Millis and C. Marianetti, arXiv:1309.2995

ICAM/BU 2013

Collaborators

M-J. Han (CU->ANL->KAIST)X. Wang (CU->U-Md)L. deMedici (ESPCI)

C. Marianetti (Columbia)

*Hyowon Park (Columbia)

Hung The Dang(Columbia->Aachen)

Ara Go(Columbia)

General Approach to Many-Body Electronic Structure

•=>partition d.o.f. into ``correlated subspace’’ (active space) and ``background’’

• treat correlated subspace by many-body method; treat background by mean field method

•embed active space into background

1. background electronic structure (DFT)2. Active subspace: (atomic-like d-orbitals) 3. On-site intra-d interactions (c-RPA)4. Solve active space (DMFT--single-site)5. Embed (double counting and charge self-consistency)

DFT+DMFT

The most important recent developments: I

Gabi Kotliar

Antoine Georges

Dynamical Mean Field Theory

Enables marriage of many-body and real-materials theory

Parametrize self energy in terms of small number N of functions of frequency.

parametrization function f determines ‘flavor’ of DMFT

Physical model: M (typically infinity) orbitals in Hilbert space

ΣaDMFT is self energy of a 0 (space) + 1 (time)d QFT

Σ(k,ω) =�

f(k)a ΣaDMFT(ω) a = 1...N

N->M; recover exact theory.

?Can we solve the theory with any (useful) N??Can we get reasonable results with reasonable N?

The most important recent developments: II

Philipp Werner Emanuel Gull

Continuous-Time Quantum Monte

Enables solution of DMFT equations in realistic contexts

Field started by Rubtsov, important contributions from Haule

P. Werner, A Comanac, Luca De Medici, M. Troyer, and A.Millis, PRL 97, 076405 (2006).E. Gull, P. Werner, O. Parcollet and M. Troyer, EPL 82 57003 (2008).

CT-QMCCT-QMC: ``many-body adaptive grid”

(Image from From E. Gull)

All methods involve manipulating matrices; cost ~cube of matrix size.

A model system

H = −�

ti−jc†iσcjσ + U

ni↑ni↓

Validation(cluster) dmft: approximation to many body problem: accuracy controlled by parameter N.Example:

1Σp(ω)→ Σapprox

p (ω) =�

φa(p)Σa(ω)

φa(p) = 1 if p is in the patchcontaining Ka and is 0 otherwise

Validation3D Hubbard model, n=1 U=8t=2W/3 T=0.4t

N-2/3Fuchs, Gull, et al PRL 106 030401 (2011)

E/site

Particle-hole asymmetric pseudogap in 2d Hubbard model

Emanuel Gull, Michel Ferrero, Olivier Parcollet, Antoine Georges, A J. M, Phys. Rev. B82 155101 (2010).

Particle-hole asymmetric pseudogap in 2d Hubbard model

0 0.2 0.4 0.6 0.8 11/N

Mott gap Pseudgogap

Mott gap and Pseudogap vs cluster sizeU=7t t'=-0.15t β=20/t

superconductivity and the pseudogap in the 2D Hubbard model

0 0.5 1 1.5Frequency [t]

x=0.03x=0.05x=0.06x=0.075x=0.10x=0.12

0 0.5 1 1.5Frequency [t]

x=0.03x=0.05x=0.06x=0.075x=0.10x=0.12

E. Gull, O. Parcollet and A. J. Millis, PRL 110 216406 (2013)

Summary: Hubbard model

The method works for a model system

--gives physically interesting answers

--theoretically reasonable convergence structure

Beyond model systems(DFT+DMFT)

Our recent work

*Xin Wang, M. J. Han, Luca de' Medici, Hyowon Park, C. A. Marianetti and Andrew J. Millis, Physical Review B86, 195136 (2012).*H. T. Dang and AJM Phys. Rev. B87, 155127 (2013), Phys. Rev. B87184434 (2013)*H. T. Dang, A. J. Millis and C. Marianetti, arXiv:1309.2995

Introduced by Liechtenstein, Anisimov, Georges, Kotliar....

now widely used..

an application: ferromagnetism in vanadate superlattices

Vanadate quantum well(LaVO3)n/(SrVO3)m

Luders et al Phys Rev B80 241102 (2009) and Boulay et al Phys Rev B83 125403 (2011)

LaVO3: Antiferromagnetic Mott insulator (d2)SrVO3: correlated metal (d1)

Superlattice: room T ferromagnetism!

(LaVO3)n/(SrVO3)1

Luders et al Phys Rev B80 241102 (2009)

Theoretical Questions are clear

(1) The design rule question: can theory reliably identify the circumstances under which a material will exhibit a new phase (magnetic, superconducting, insulating...) or an optimized version of an existing phase

(0) The modelling question: can theory reliably obtain the values of physical quantities--gaps, transition temperatures, charge transfers......

Experimental situation complicated

Design Rules

*Qualitative: identify important features/trends

*Quantitative: predict Tc or at least sign of Tc

Qualitative Design Rules:limitations of reasoning by analogy

High-Tc copper-oxide superconductivity

*One band*Proximity to Mott phase*S=1/2

High-Tc iron-pnictide superconductivity

*Several bands*Proximity to metallic SDW*`Hunds metal’ (local spin S>1/2)

Theory needed, even for qualitative trends

*Theory needed*Density functional band theory (at least in its present implementation) inadequate*Complete solution of all-electron correlation problem is impossible

=> need to (1) Identify subset of orbitals to be correlated(2) Solve correlation problem (3) Embed solution in wider electronic structure.

DFT+DMFT La/SrVO3

Ferromagnetism favored by large tilts, distance from AF Mott insulator

Key feature: rotations of VO3 octahedron

Bulk La/SrVO3

DFT+DMFT La/SrVO3

Bulk La/SrVO3

In LVO/SVO solid solutions, doping away from Mott insulator also moves tilt angle in wrong way!

What about superlattices?

DFT+DMFT La/SrVO3

Bulk La/SrVO3 In a superlattice, carrier concentration and tilt may be independently controlled=>? move into magnetic regime?

DFT+DMFT La/SrVO3

!!Superlattice moves tilts in the wrong way!!

Bulk La/SrVO3 Superlattice

DFT+DMFT La/SrVO3

!!Superlattice moves tilts in the wrong way!!

Bulk La/SrVO3 Superlattice

Unsatisfactory aspect of theorymany moving parts

=>?are we doing the right thing?

1. background electronic structure (DFT)2. Active subspace: (atomic-like d-orbitals) 3. On-site intra-d interactions (c-RPA)4. Solve active space (DMFT--single-site)5. Embed (double counting and charge self-consistency)

DFT+DMFT

the loop

VKohn−Sham(r)ρ(r) G0(r, r�;ω)

Σ̂ = |d > Σ(ω) < d|�G0(r, r�;ω)−1 − Σ̂(ω)

�−1G(r, r�;ω) =

ρ(r) =�

πf(ω)G(r, r;ω)

Double counting shift

Charge self consistency

Onsite U, J pre-computed

We tried this out

Material Expt Th

We tried this out

Material Expt Th

We tried this out

Material Expt Th

SrVO3 M

We tried this out

Material Expt Th

SrVO3 M M

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I M

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I M

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I M

LaVO3 I

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I M

LaVO3 I M

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I M

LaVO3 I M

La2CuO4

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I M

LaVO3 I M

La2CuO4 I

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I M

LaVO3 I M

La2CuO4 I M

We tried this out

Material Expt Th

SrVO3 M M

LaTiO3 I M

LaVO3 I M

La2CuO4 I M

?What went wrong?

Possibilities

•Wrongly chosen active space (need to treat more than just on-site d-d interactions dynamically)

•Wrong embedding (double counting)

•Wrong approximation (need more than single-site DMFT)

•Wrong treatment of “background” electrons

Our view now

•Pseudo cubic materials: the problem is with the underlying band structure

•Cuprates: need to fix underlying band structure and go beyond single-site DMFT

Crucial Importance of p-d energy splitting

SrVO3: an instructive example

SrVO3: an instructive exampleCubic, moderately correlated metal

DFT+DMFT TRIQS Code

process keeps relative separation of p and d bands independent of U--and slightly smaller than band theory value

Without charge self consistency, p-bands stay at DFT position

DFT+DMFT TRIQS Code

The Zaanen-Sawatzky-Allen phase diagram (integer band filling)

εd − εp

UCharge transfer Insulator

Mott Insulator

εd − εp

Mott Insulator

Calculation says that increasing U moves you more or less vertically in the ZSA phase diagram.

εd − εp

Mott Insulator

Calculation says that increasing U moves you more or less vertically in the ZSA phase diagram.

If the material starts with too small a p-d energy splitting, it remains in the metallic regime as U increases

Would like to explore all of ZSa phase diagram

Fully charge self consistent calculation explores only one line.

?how to generalize?

d-occupancy:* Intuitive notion: e.g. La3+Ti3+O32- =>Ti d1 (Nd=1)

*Theoretically needed (if you want to put correlations on d-orbital you need to know what this orbital is and how much it is occupied)

*definition:In terms of exact Green function G(r, r

�;ω)

and predefined d-wave function φd

Nd =�

�dω

πf(ω)

�d3rd3r

�(φa

d(r))∗ Gσ(r, r�,ω)φa

d(r�)�

*d occupancy depends on how orbital is defined (as does the entire edifice of DFT+DMFT)

We have found: all reasonable definitions give consistent answers

SrVO3:

lower panels: ad-hoc double counting correction chosen to keep Nd at value found in fully charge self consistent calculation.

DFT+DMFT TRIQS Code

SrVO3:

Spectra are identical.

DFT+DMFT TRIQS Code

SrVO3:

Spectra are identical.

DFT+DMFT TRIQS Code

=>dont need to bother with charge self consistency; plot results in terms of Nd

Metal-insulator phase diagrams

Titanates/Vanadates

For each U, adjust Nd so theory gives insulating gap in agreement with experiment. Compute oxygen bands.

Titanates/Vanadates

For each U, adjust Nd so theory gives insulating gap in agreement with experiment. Compute oxygen bands. Relatively narrow U-range consistent with expt

Titanates/Vanadates

For each U, adjust Nd so theory gives insulating gap in agreement with experiment. Compute oxygen bands. Relatively narrow U-range consistent with expt

U~5-6eV is found in C-RPA calculations

Possibilities

•Wrongly chosen active space (need to treat more than just on-site d-d interactions dynamically)

•Wrong embedding (double counting)

•Wrong approximation (need more than single-site DMFT)

•Wrong treatment of “background” electrons

Summary: titanates/vanadates

Single-site DMFT is acceptable first-order approximation to electronic structure **IF oxygen bands are suitably positioned**

Fully charge self consistent DFT+DMFT places oxygen bands higher than DFT and thus too high relative to experiment.

=>?need better background electronic structure?

Cuprates: beyond single-site dmft Ara Go. (New CI-based ``impurity solver’’)

! [eV]

Nd=1 Nd=4

2 4 6 8 10 12 14 16

InsulatorMetal

MarginalCoexistence

1.1 1.2 1.3 1.4 1.5 1.6

! [eV]

Nd=1 Nd=4

2 4 6 8 10 12 14 16

InsulatorMetal

MarginalCoexistence

1.1 1.2 1.3 1.4 1.5 1.6

Nd=1, U=8.0eVNd=4, U=8.0eVNd=1, U=9.6eVNd=4, U=9.6eVNd=1, U=19.2eVNd=4, U=19.2eV

Resolves old problem in optics

Xin Wang2011

Ara Go: 2013

Summary: cuprates

4-site DMFT is acceptable first-order approximation to electronic structure **IF oxygen bands are suitably positioned**

Long-standing problem with optics resolved

Very recent success:energetics of nontrivial phases

metal-insulator transition in rare-earth nickel oxides

Metal-insulator transition occurs along with 2 sublattice Ni-O breathing distortion

J A Alonso et al PRL 82 3871 (1999)

Wrongly interpreted as charge order

DFT+DMFT using observed structure of insulating phase

Hyowon Park, Andrew J. Millis, and Chris A. Marianetti, PRL 109, 156402 (2012)

N1=8.24 N2=8.22

Find: insulator but no charge order

Site selective Mott transition

Can now compute energiesand pressure-volume phase diagrams

Hyowon Park, Andrew J. Millis, and Chris A. Marianetti, to appear

Conclusion

Conclusion: need underlying electronic structure that gets the oxygen energies right. DFt puts O states too close to the fermi level

1. background electronic structure 1.1. beyond DFT method

2. Active subspace: ?atomic-like d-orbitals? 3. ?On-site intra-d interactions?4. Solve active space (?DMFT?)

4.1. if only on-site interactions4.2. is single-site approx ok?

5. Embed

Prospects: DFT+DMFT

Cuprates

U-Nd phase diagramDFT+DMFT: treat only Cu-d x2-y2 orbital dynamicallyNon-self-consistent

band theory

‘distance’ from band theory Nd to metal-insulator phase boundary ~0.25el/Cu

U-Nd phase diagramDFT+DMFT: treat Cu-d x2-y2 orbital dynamicallyNon-self-consistent

VASP/MLWFWein2K/Projector

More detailed lookTreat x2-y2 dynamically, all other d with Hartree-Fock J=0 vs J=0.7eV

treat all 5 d dynamically (only Ising terms) J=0 vs J=0.7eV

What is band theory Nd?

VASP/Wannier

Wein2K/projector

What about full charge self-consistency

Wein2K/projector

?How to position the materials?

DFT+DMFT and its Discontents: towards a first-principles theory of correlated electron...

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